The distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)` then equation of the hyperbola is

Options

A. `x^(2)-y^(2)=32`
B. `(x^(2))/4-(y^(2))/9=1`
C. `2x-3y^(2)=7`
D. none of these

Explanation

To find the equation of the hyperbola given the distance between the foci and the eccentricity, we can follow these steps:

### Step 1: Understand the relationship between the distance between the foci and 'a'
The distance between the foci of a hyperbola is given by the formula \(2c\), where \(c\) is the distance from the center to each focus. Given that the distance between the foci is 16, we can write:
\[ 2c = 16 \implies c = 8 \]

**Hint:** Remember that the distance between the foci is always \(2c\).

### Step 2: Use the relationship between eccentricity, 'a', and 'c'
The eccentricity \(e\) of a hyperbola is defined as:
\[ e = \frac{c}{a} \]
Given that \(e = \sqrt{2}\), we can substitute \(c\) to find \(a\):
\[ \sqrt{2} = \frac{8}{a} \implies a = \frac{8}{\sqrt{2}} = 4\sqrt{2} \]

**Hint:** The eccentricity relates the distance to the foci and the distance to the vertices.

### Step 3: Find \(b^2\) using the relationship \(c^2 = a^2 + b^2\)
We know that:
\[ c^2 = a^2 + b^2 \]
Substituting the values we have:
\[ 8^2 = (4\sqrt{2})^2 + b^2 \]
Calculating \(c^2\) and \(a^2\):
\[ 64 = 32 + b^2 \]
Solving for \(b^2\):
\[ b^2 = 64 - 32 = 32 \]

**Hint:** This relationship helps us find \(b\) once we have \(c\) and \(a\).

### Step 4: Write the standard form of the hyperbola
The standard equation of a hyperbola centered at the origin is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting \(a^2 = 32\) and \(b^2 = 32\):
\[ \frac{x^2}{32} - \frac{y^2}{32} = 1 \]

### Step 5: Simplify the equation
We can simplify this equation:
\[ \frac{x^2 - y^2}{32} = 1 \implies x^2 - y^2 = 32 \]

**Hint:** Always simplify the equation to its standard form.

### Final Result
Thus, the equation of the hyperbola is:
\[ \boxed{x^2 - y^2 = 32} \]